We consider the attractive Bose-Einstein condensate (BEC) of Rb atoms, which is confined in a harmonic plus quartic trap $[V(r ) = ½m^2r^2 + ${\lambda}$r^4]$. We slowly tune the coefficient of quartic term (${\lambda}$) to see how the metastable region (MSR) in the outer part of the trap is modified. The MSR is always bounded by aleft intermediate barrier (IB) and a deep but narrow attractive well (NAW) appears on the left of this barrier. For ${\lambda}< 0$, the MSR is bounded by an additional right IB. A dramatic behaviour (compared with the attractive BEC in a harmonic trap) of interaction energy, kinetic energy, trap energy along with the total ground-state energy of the condensate is observed in this different metastability. For $\lambda> 0$, the outer wall of the MSR increases steeply as ${\lambda}$ increases. Dynamical responses of the zero point energies of the condensate are explored also. Observations areinteresting as we find a critical value of anharmonicity (${\lambda}_cr$) to collapse the condensate. We slightly increase the strength of ${\lambda}$ (close to ${\lambda}_cr$) and investigate the significant increase of the quantum tunnelling probability throughthe left IB into an inner core (NAW).